Method and apparatus for modeling and estimating the characteristics of a power amplifier by retaining even-order terms in estimating characteristics

ABSTRACT

Disclosed is an apparatus and method for modeling and estimating the characteristics of a power amplifier. A predistortion module generates a predistorted signal in response to a predistortion function and an input signal. A power amplifier receives the predistorted signal and generates an output signal. A polynomial module generates coefficients of a complex polynomial of order p (p is an integer greater than one) in response to the predistorted signal and the output signal. In particular, the complex polynomial is inplemented with both even and odd terms, thereby improving the ability to accurately model the power amplifier.

RELATED APPLICATIONS

[0001] The present patent application is related to U.S. patentapplication Ser. No. ______ KIM 7______, entitled, “METHOD AND APPARATUSFOR MODELING AND ESTIMATING THE CHARACTERISTICS OF A POWER AMPLIFIERWITH MEMORY”, being concurrently filed herewith and having a filing dateof ______; to U.S. patent application Ser. No. ______ KIM 8-4______,entitled, “METHOD AND APPARATUS FOR CALCULATING THE PREDISTORTIONFUNCTION FROM A POWER AMPLIFIER”, being concurrently filed herewith andhaving a filing date of ______; and to U.S. patent application Ser. No.______ KIM 9-5______, entitled, “METHOD AND APPARATUS FOR CALCULATINGTHE PREDISTORTION FUNCTION FROM A POWER AMPLIFIER”, being concurrentlyfiled herewith and having a filing date of ______; all of which have acommon inventor and assignee and being incorporated herein by reference.

FIELD OF THE INVENTION

[0002] This invention relates generally to signal amplification and, inparticular, to determining signal amplifier characteristics forintentionally induced distortion techniques utilized prior to and inconjunction with signal amplification.

BACKGROUND OF THE INVENTION

[0003] In the field of radio communication systems, it is a well-knownproblem that the power amplifiers present in transmission equipmentoperate in a non-linear fashion when the power amplifiers are operatednear their peak output. As a result, the power amplifier introducessignificant signal distortion that can appear in various forms. Forexample, if more than one signal is input into the power amplifier orpower amplification stage, its non-linear characteristics can cause anundesirable multiplicative interaction of the signals being amplified,and the power amplifier's output can contain intermodulation products.These intermodulation products cause interference and crosstalk over thepower amplifier's operational frequency range.

[0004] In power amplifier design, there is a trade off betweendistortion performance and efficiency. Linear amplifiers that operateunder “Class A” conditions create little distortion but are inefficient,whereas nonlinear amplifiers operated under “Class C” conditions arereasonably efficient but introduce significant distortions. While bothefficiency and distortion are important considerations in amplifierdesign, efficiency becomes increasingly important at high power levels.Because of their efficiency, nonlinear amplifiers are largely preferred,leaving the user with the problem of distortion.

[0005] In order to employ nonlinear power amplifiers, techniques havebeen used to improve linearity and thereby reduce the effects ofinterference and crosstalk. Linearity can be achieved by application ofvarious linearization techniques that reduce the distortion caused bynonlinear amplification. Conventional amplifier linearization techniquescan be broadly categorized as feedback, feedforward, or predistortion.

[0006] The last mentioned technique, predistortion, intentionallydistorts the signal before the power amplifier so that the non-linearityof the power amplifier can be compensated. According to this technique,linearization is achieved by distorting an input signal according to apredistortion function in a manner that is inverse to the amplifiercharacteristic function. The predistortion technique can be applied atradio frequency (RF), intermediate frequency (IF), or at baseband.

[0007] In the baseband domain, the input signal information is at a muchlower frequency, allowing digital methods to be employed. Thepredistortion function is applied to the input signal with the resultingpredistorted signal being upconverted to IF and then finally to the RFcarrier frequency. It is also possible to apply adaptive predistortiontechniques where feedback from the output of the amplifier is used toupdate and correct the predistortion function.

[0008] The form of the predistortion function is dependent upon themodel used to characterize the output of the amplifier. Predistortionfunctions in the baseband domain are typically implemented as a table ofgain and phase weighting values within a digital signal processor. ACartesian feedback method employs a quadrature representation of thesignal being amplified. The incoming quadrature signals I and Q arecompared to the feedback quadrature signals. Thus, there are two sets ofcoefficients, one for each quadrature channel, that are being updated tomodel the predistortion characteristics. In this manner, gain and phasenon-linearities within the amplifier can be compensated. Performance isdependent on the size of the look up table and the number of bits usedto represent the signal. Better performance and more adaptivity isachieved with larger look up tables and more bits albeit at the expenseof longer processing times.

[0009] Predistortion functions are also modeled as polynomials. Idealamplifiers have linear characteristics; consequently, amplifiers withslight non-linearities can be modeled as polynomials of only a fewterms, with only odd terms being employed. Even terms are discardedbecause their use with negative-valued inputs can interfere withlinearity. While processing demands are eased by excluding and limitingthe number of terms in the polynomial modeling, performance issacrificed.

[0010] Accordingly, there is a need for a device to more quickly andefficiently determine the characteristics of a non-linear amplifier.

SUMMARY OF THE INVENTION

[0011] The present invention teaches an apparatus and method formodeling and estimating the characteristics of a power amplifier. Apredistortion module generates a predistorted signal in response to apredistortion function and an input signal. A power amplifier receivesthe predistorted signal and generates an output signal. A polynomialmodule generates coefficients of a complex polynomial of order p (p isan integer greater than one) in response to the predistorted signal andthe output signal. The coefficients characterize the power amplifier.The complex polynomial is implemented with both even and odd terms. Evenorder terms are typically ignored for the polynomial modeling ofamplifiers because the non-linear distortion signals caused by poweramplifiers are strongest at odd order harmonic frequencies.Additionally, it is commonly thought that the use of even terms withnegative valued inputs can interfere with the linearity, sincenegativity is lost at even powers. However, the inclusion of even orderterms improves the ability to accurately model the power amplifier andthereby improves predistortion performance.

[0012] In another exemplary embodiment of the present device, thepolynomial module employs a minimum mean squared error criteria todetermine said polynomial coefficients, thereby allowing a very fast andefficient implementation.

[0013] By improving the ability to model power amplifiers, the presentinvention improves the ability to model the power amplifierpredistortion function. The invention further enables power amplifiersto be operated in the nonlinear region near saturation, yet suppressesundesirable intermodulation products. Resort to a larger amplifier, tokeep operation within the linear region, is avoided. Power amplifiersizes are kept small with associated cost savings, particularlyimportant in the field of wireless communications.

[0014] The above factors make the present invention essential foreffective power amplifier predistortion.

BRIEF DESCRIPTION OF THE DRAWINGS

[0015] For a better understanding of the present invention, referencemay be had to the following description of exemplary embodimentsthereof, considered in conjunction with the accompanying drawings, inwhich:

[0016]FIG. 1 is a block diagram providing an overview of an exemplarysystem employing adaptive power amplifier predistortion;

[0017]FIG. 2 is a block diagram of the simplified baseband model forpower amplifier predistortion; and

[0018]FIG. 3 is a block diagram illustrating the device of FIG. 2 asused in a RF transmission system in accordance with the principles ofthe present invention.

DETAILED DESCRIPTION

[0019] The following description is presented to enable a person skilledin the art to make and use the invention, and is provided in the contextof a particular application and its requirements. Various modificationsto the disclosed embodiments will be readily apparent to those skilledin the art, and the general principles defined herein may be applied toother embodiments and applications without departing from the spirit andthe scope of the invention. Thus, the present invention is not intendedto be limited to the embodiments disclosed, but is to be accorded thewidest scope consistent with the principles and features disclosedherein.

[0020] The specification initially discusses the general concept andprinciples of adaptive digital predistortion in view of the novel systemfor determining the characteristics of a power amplifier. Exemplaryembodiments of the system for determining the characteristics of a poweramplifier are then described.

[0021] Overview of Adaptive Power Amplifier Predistortion

[0022] The principal benefit of the present invention is the ability tomore efficiently model the power amplifier characteristics in order toimprove the ability to employ adaptive digital predistortion (ADPD). Thestructure of an exemplary ADPD system is seen in FIG. 1. An initialbaseband digital signal 10 is identified as u_(n), where n is the timeindex. The initial baseband digital signal 10 is fed into apredistortion system 20 that is described as a function A(•). The outputof the predistortion system 20 is the baseband digital input signal 12to the power amplifier 50 and is defined as x_(n). The baseband digitalinput signal 12 is processed by a digital to analog (D/A) converter 30with the resulting baseband analog signal being processed by anup-conversion means 40 that is comprised of mixers and filters andoperates in the intermediate frequency (IF) range. The up-conversionmeans 40 outputs a signal in the radio frequency (RF) range and feedsthe signal to the power amplifier 50. While there are many methods forADPD, the approach with the exemplary invention can be divided into twosteps. First, the characteristics of the power amplifier 50 areestimated. Then, the predistortion function based on the poweramplifier's 50 characteristics is obtained. For proper characterizationof the power amplifier 50, the time domain inputs and outputs of thepower amplifier 50 need to be compared. Thus, the output of the poweramplifier 50 is tapped and fed back to a down-conversion means 70. Likethe up-conversion means 40, the down-conversion means 70 requires mixersand filters in the IF range. The output of the down-conversion means 70is fed into an analog to digital (A/D) converter 80. The output of theA/D converter 80 is input into a means for delay adjustment 82 with itsoutput representing the baseband digital output signal 14 identified asy_(n). The baseband digital output signal 14 and the baseband digitalinput signal 12 are input to the polynomial module 25 in order todetermine the coefficients that characterize the power amplifier 50. Theoutput of the polynomial module 25 is coupled to the predistortionpolynomial module 15 that determines the predistortion coefficients thatare fed into the predistortion module 20. The polynomial module,predistortion polynomial module and predistortion module may beimplemented in hardware, or in other forms such as software or firmware.

[0023] As implemented in FIG. 1, the baseband digital input signal 12 tothe power amplifier 50 as well as the baseband digital output signal 14of the power amplifier 50 are easily accessible. However, theup-conversion means 40 and the down-conversion means 70 distort thesignals. Mixers are nonlinear devices and will add non-lineardistortions. Furthermore, the phase responses of analog filters are notlinear, thereby causing different time delays for different frequencycomponents. Generally, these distortions can be considered negligible orcan be corrected by using linear filters, and considered to be part ofthe baseband model for the power amplifier.

[0024] By neglecting the effect of up-conversion and down-conversionprocess, the whole predistortion process can be considered in thebaseband domain. In FIG. 2, an exemplary baseband model forpredistortion processing is illustrated. The power amplifier 150 isdefined as a baseband function B(.) with complex inputs and complexoutputs.

[0025] Polynomial Modeling of the Power Amplifier

[0026] Predistortion requires the information on the characteristics ofthe power amplifier 150. Proper baseband modeling of the power amplifier150 is described herein. Since the power amplifier 150 is operating inradio frequency (RF) domain, the baseband model of the power amplifier150 must be considered in complex numbers. Letting x and y, aspreviously defined, be the input and output of the power amplifier 150,the following relationships can be defined, $\begin{matrix}{\begin{matrix}{y = \quad {B\left( {\overset{\rightarrow}{b},x} \right)}} \\{= \quad \left. {{b_{1}x} + b_{2}} \middle| x \middle| {x + b_{3}} \middle| x \middle| {}_{2}{x + \ldots + b_{p}} \middle| x \middle| {}_{p - 1}x \right.}\end{matrix}\quad} & (1)\end{matrix}$

[0027] where p is the order of the polynomial, b_(k)=b_(kr)+jb_(ki),k∈{1, 2, . . . , p}and {right arrow over (b)} is a size 2p vector ofcomplex polynomial coefficients defined as [b_(1r), b_(2r), . . . ,b_(pr), b_(1i), b_(2i), . . . , b_(pi)]. In general, p=5 is sufficientto model the power amplifier 150.

[0028] As seen in equation (1), the complex polynomial is implementedwith both even and odd terms, thereby improving the ability toaccurately model the power amplifier. Typically, even order terms areignored for polynomial modeling because the non-linear distortionsignals caused by power amplifier are located at odd order harmonicfrequencies. Moreover, the use of even order terms is generally bypassedbecause it is thought that negative-valued inputs can interfere withlinearity. However, inclusion of even order terms as seen in the presentinvention retains the negativity of those terms and allows for bettermodeling of the power amplifier, thereby improving the predistortionperformance.

[0029] Estimation of Power Amplifier Characteristics

[0030] An estimation is performed in order to obtain an optimum {rightarrow over (b)} that describes the characteristics of the poweramplifier 150. A minimum mean squared error criteria is employed basedupon the complex input and output samples of the power amplifier 150.x_(n) is the input sample and y_(n) is the corresponding output sample,where n ∈{1, 2, . . . , N}. The error function f({right arrow over (b)})is defined as $\begin{matrix}{\begin{matrix}{{f\left( \overset{\rightarrow}{b} \right)} = \quad {E\left\lbrack \left| {y_{n} - {B\left( {\overset{\rightarrow}{b},x_{n}} \right)}} \right|^{2} \right\rbrack}} \\{= \quad \left. {\frac{1}{N}\sum\limits_{n = 1}^{N}} \middle| {y_{n} - {B\left( {\overset{\rightarrow}{b},x_{n}} \right)}} \right|^{2}} \\{= \quad {\frac{1}{N}{\sum\limits_{n = 1}^{N}\left( \left| y_{n} \middle| {}_{2}{{{- y_{n}^{*}}{B\left( {\overset{\rightarrow}{b},x_{n}} \right)}} - {y_{n} \cdot}} \right. \right.}}} \\{\quad \left. \left. {{B\left( {\overset{\rightarrow}{b},x_{n}} \right)}* +} \middle| {B\left( {\overset{\rightarrow}{b},x_{n}} \right)} \right|^{2} \right)}\end{matrix}\quad} & (2)\end{matrix}$

[0031] where E[x] is the mean of x, and x* is the complex conjugate ofx. Minimizing the error or merit function results in the most accuratemodeling of the power amplifier 150 and thereby the optimal polynomialcoefficients. From equations (1) and (2), it is known that f({rightarrow over (b)}) is a quadratic function of {right arrow over (b)}.Thus, f({right arrow over (b)}) can be expressed by a Taylor series as$\begin{matrix}{{f\left( {\overset{\rightarrow}{b} + \overset{\rightarrow}{d}} \right)} = {{f\left( \overset{\rightarrow}{b} \right)} + {{\nabla{f\left( \overset{\rightarrow}{b} \right)}} \cdot \overset{\rightarrow}{d^{\quad t}}} + {\frac{1}{2}{\overset{\rightarrow}{d} \cdot {H\left( \overset{\rightarrow}{b} \right)} \cdot \overset{\rightarrow}{d^{\quad t}}}}}} & (3)\end{matrix}$

[0032] Where “t” is the transpose of the matrix, and ∇f({right arrowover (b)}) is the gradient of f({right arrow over (b)}), defined as$\begin{matrix}{{\nabla{f\left( \overset{\rightarrow}{b} \right)}} \equiv \left\lbrack {\frac{\partial f}{\partial b_{1r}},\frac{\partial f}{\partial b_{2r}},\ldots \quad,\frac{\partial f}{\partial b_{p\quad r}},\frac{\partial f}{\partial b_{1i}},\frac{\partial f}{\partial b_{2i}},\ldots \quad,\frac{\partial f}{\partial b_{p\quad i}}} \right\rbrack} & (4)\end{matrix}$

[0033] H({right arrow over (b)}) is the Hessian or second orderderivative of f({right arrow over (b)}), and is defined as$\begin{matrix}{{H\left( \overset{\rightarrow}{b} \right)} \equiv \begin{bmatrix}\frac{\partial^{2}f}{\partial b_{1r}^{2}} & \frac{\partial^{2}f}{{\partial b_{1r}}{\partial b_{2r}}} & \ldots & \frac{\partial^{2}f}{{\partial b_{1r}}{\partial b_{p\quad r}}} & \frac{\partial^{2}f}{{\partial b_{1r}}{\partial b_{1i}}} & \frac{\partial^{2}f}{{\partial b_{1r}}{\partial b_{2i}}} & \ldots & \frac{\partial^{2}f}{{\partial b_{1r}}{\partial b_{p\quad i}}} \\\frac{\partial^{2}f}{\partial b_{2r}^{2}} & \frac{\partial^{2}f}{{\partial b_{2r}}{\partial b_{2r}}} & \ldots & \frac{\partial^{2}f}{{\partial b_{2r}}{\partial b_{p\quad r}}} & \frac{\partial^{2}f}{{\partial b_{2r}}{\partial b_{1i}}} & \frac{\partial^{2}f}{{\partial b_{2r}}{\partial b_{2i}}} & \ldots & \frac{\partial^{2}f}{{\partial b_{2r}}{\partial b_{p\quad i}}} \\\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\\frac{\partial^{2}f}{\partial b_{p\quad r}^{2}} & \frac{\partial^{2}f}{{\partial b_{p\quad r}}{\partial b_{2r}}} & \ldots & \frac{\partial^{2}f}{{\partial b_{p\quad r}}{\partial b_{p\quad r}}} & \frac{\partial^{2}f}{{\partial b_{p\quad r}}{\partial b_{1i}}} & \frac{\partial^{2}f}{{\partial b_{p\quad r}}{\partial b_{2i}}} & \ldots & \frac{\partial^{2}f}{{\partial b_{p\quad r}}{\partial b_{p\quad i}}} \\\frac{\partial^{2}f}{\partial b_{1i}^{2}} & \frac{\partial^{2}f}{{\partial b_{1i}}{\partial b_{2r}}} & \ldots & \frac{\partial^{2}f}{{\partial b_{1i}}{\partial b_{p\quad r}}} & \frac{\partial^{2}f}{{\partial b_{1i}}{\partial b_{1i}}} & \frac{\partial^{2}f}{{\partial b_{1i}}{\partial b_{2i}}} & \ldots & \frac{\partial^{2}f}{{\partial b_{1i}}{\partial b_{p\quad i}}} \\\frac{\partial^{2}f}{\partial b_{2i}^{2}} & \frac{\partial^{2}f}{{\partial b_{2i}}{\partial b_{2r}}} & \ldots & \frac{\partial^{2}f}{{\partial b_{2i}}{\partial b_{p\quad r}}} & \frac{\partial^{2}f}{{\partial b_{2i}}{\partial b_{1i}}} & \frac{\partial^{2}f}{{\partial b_{2i}}{\partial b_{2i}}} & \ldots & \frac{\partial^{2}f}{{\partial b_{2i}}{\partial b_{p\quad i}}} \\\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\\frac{\partial^{2}f}{\partial b_{p\quad i}^{2}} & \frac{\partial^{2}f}{{\partial b_{p\quad i}}{\partial b_{2r}}} & \ldots & \frac{\partial^{2}f}{{\partial b_{p\quad i}}{\partial b_{p\quad r}}} & \frac{\partial^{2}f}{{\partial b_{p\quad i}}{\partial b_{1i}}} & \frac{\partial^{2}f}{{\partial b_{p\quad i}}{\partial b_{2i}}} & \ldots & \frac{\partial^{2}f}{{\partial b_{p\quad i}}{\partial b_{p\quad i}}}\end{bmatrix}} & (5)\end{matrix}$

[0034] Since f({right arrow over (b)}) is a quadratic function, it has aglobal minimum if H({right arrow over (b)}) is a positive definitematrix. By the same reason, the Hessian is not the function of {rightarrow over (b)} and the notation of H can be used instead of H({rightarrow over (b)}). The estimation obtains the optimum {right arrow over(b_(opt))} that makes f({right arrow over (b_(opt))}) on its minimum.Using the Newton Method, the optimum {right arrow over (b_(opt))} can beobtained. Letting {right arrow over (b₀)} be an initial value, then{right arrow over (b_(opt))} can be expressed as

{right arrow over (b_(opt))}= {right arrow over (b₀)}+ {right arrow over(d)}  (6)

[0035] Equation (3) can then be re-written as $\begin{matrix}{{f\left( \overset{\rightarrow}{b_{o\quad p\quad t}} \right)} = {{f\left( {\overset{\rightarrow}{b_{0}} + \overset{\rightarrow}{d}} \right)} = {{f\left( \overset{\rightarrow}{b_{0}} \right)} + {{\nabla{f\left( \overset{\rightarrow}{b_{0}} \right)}} \cdot \overset{\rightarrow}{d^{\quad t}}} + {\frac{1}{2}{\overset{\rightarrow}{d} \cdot H \cdot \overset{\rightarrow}{d^{\quad t}}}}}}} & (7)\end{matrix}$

[0036] If H is a positive definite matrix, the optimum {right arrow over(d)} is found by making the gradient of f({right arrow over (b₀)}+{rightarrow over (d)}) based on {right arrow over (d)} equal zero.

{right arrow over (d)}=−H ⁻¹ ·∇f({right arrow over (b₀)})  (8)

[0037] Thus, the optimum polynomial coefficients {right arrow over(b_(opt))} is

{right arrow over (b_(opt))}= {right arrow over (b₀)}− H ⁻¹ ·∇f({rightarrow over (b₀)})  (9)

[0038] The gradient of f({right arrow over (b₀)}) is calculated fromequations (1), (2), and (4). $\begin{matrix}{{\nabla{f\left( \overset{\rightarrow}{b_{0}} \right)}} = {\nabla\left( \left. {\frac{1}{N}\sum\limits_{n = 1}^{N}} \middle| {y_{n} - {B\left( {\overset{\rightarrow}{b_{o}},x_{n}} \right)}} \right|^{2} \right)}} & (10)\end{matrix}$

[0039] Where R[x] and I[x] are the real components of x and theimaginary component of x, respectively. From the results,$\begin{matrix}{{{\nabla{B\left( {\overset{\rightarrow}{b_{o}},x_{n}} \right)}} = \left\lbrack {x_{n},\left| x_{n} \middle| x_{n} \right.,\ldots \quad,\left| x_{n} \middle| {}_{p - 1}x_{n} \right.,{j\quad x_{n}},\left. j \middle| x_{n} \middle| x_{n} \right.,\left. {\ldots \quad j} \middle| x_{n} \middle| {}_{p - 1}x_{n} \right.} \right\rbrack}{{t\quad h\quad e\quad n},}} & (11) \\{{{\nabla f}\left( \overset{\rightarrow}{b_{0}} \right)} = \begin{bmatrix}{{b_{o1r}M_{2}} + {b_{o2r}M_{3}} + \ldots + {b_{o\quad p\quad r}M_{p + 1}} - {\left\lbrack C_{1} \right\rbrack}} \\{{b_{o1r}M_{3}} + {b_{o2r}M_{4}} + \ldots + {b_{o\quad p\quad r}M_{p + 2}} - {\left\lbrack C_{2} \right\rbrack}} \\\vdots \\{{b_{o1r}M_{p + 1}} + {b_{o2r}M_{p + 2}} + \ldots + {b_{o\quad p\quad r}M_{2p}} - {\left\lbrack C_{p} \right\rbrack}} \\{{b_{o1i}M_{2}} + {b_{o2i}M_{3}} + \ldots + {b_{o\quad p\quad i}M_{p + 1}} - {\mathcal{I}\left\lbrack C_{1} \right\rbrack}} \\{{b_{o1i}M_{3}} + {b_{o2i}M_{4}} + \ldots + {b_{o\quad p\quad i}M_{p + 2}} - {\mathcal{I}\left\lbrack C_{2} \right\rbrack}} \\\vdots \\{{b_{o1i}M_{p + 1}} + {b_{o2i}M_{p + 2}} + \ldots + {b_{o\quad p\quad i}M_{2p}} - {\mathcal{I}\left\lbrack C_{p} \right\rbrack}}\end{bmatrix}} & (12) \\{\quad {w\quad h\quad e\quad r\quad e}} & \quad \\\left. {M_{k} \equiv {\frac{2}{N}\sum\limits_{n = 1}^{N}}} \middle| x_{n} \right|^{k} & (13) \\\left. {C_{k} \equiv {\frac{2}{N}\sum\limits_{n = 1}^{N}}} \middle| x_{n} \middle| {}_{k - 1}{{\cdot x_{n}} \cdot y_{n}^{*}} \right. & (14)\end{matrix}$

[0040] Thus, the Hessian of f({right arrow over (b₀)}) $\begin{matrix}{H = {\frac{2}{N}\quad {\sum\limits_{n = 1}^{N}\begin{bmatrix}\left| x_{n} \right|^{2} & \left| x_{n} \right|^{3} & \ldots & \left| x_{n} \right|^{p + 1} & 0 & 0 & \ldots & 0 \\\left| x_{n} \right|^{3} & \left| x_{n} \right|^{4} & \ldots & \left| x_{n} \right|^{p + 2} & 0 & 0 & \ldots & 0 \\\vdots & \vdots & \ldots & \vdots & \vdots & \vdots & \ldots & \vdots \\\left| x_{n} \right|^{p + 1} & \left| x_{n} \right|^{p + 2} & \ldots & \left| x_{n} \right|^{2p} & 0 & 0 & \ldots & 0 \\0 & 0 & \ldots & 0 & \left| x_{n} \right|^{2} & \left| x_{n} \right|^{3} & \ldots & \left| x_{n} \right|^{p + 1} \\0 & 0 & \ldots & 0 & \left| x_{n} \right|^{3} & \left| x_{n} \right|^{4} & \ldots & \left| x_{n} \right|^{p + 2} \\\vdots & \vdots & \ldots & \vdots & \vdots & \vdots & \ldots & \vdots \\0 & 0 & \ldots & 0 & \left| x_{n} \right|^{p + 1} & \left| x_{n} \right|^{p + 2} & \ldots & \left| x_{n} \right|^{2p}\end{bmatrix}}}} & (15)\end{matrix}$

[0041] Defining a p×p matrix K as $\begin{matrix}{K \equiv \begin{bmatrix}M_{2} & M_{3} & \ldots & M_{p + 1} \\M_{3} & M_{4} & \ldots & M_{p + 2} \\\vdots & \vdots & \ldots & \vdots \\M_{p + 1} & M_{p + 2} & \ldots & M_{2p}\end{bmatrix}} & (16)\end{matrix}$

[0042] The Hessian can now be expressed in terms of K. $\begin{matrix}{H = \begin{bmatrix}K & 0 \\0 & K\end{bmatrix}} & (17)\end{matrix}$

[0043] Where 0 is a zero matrix of size p×p. Finally, the InverseHessian can be obtained as $\begin{matrix}{H^{- 1} = \begin{bmatrix}K^{- 1} & 0 \\0 & K^{- 1}\end{bmatrix}} & (18)\end{matrix}$

[0044]FIG. 3 illustrates a base station 310 with power amplifiersemploying a predistortion linearization technique in accordance with theprinciples of the present invention.

[0045] As shown in FIG. 3, base station 310 comprises a pair oftransmitters 320 each having a power amplifier as shown in FIG. 2. Basestation 310 can comprise, if necessary, a single transmitter oradditional transmitters. In addition, base station 310 includes asuitable transmit antenna 315 for transmission in a RF transmissionsystem that comprises both wireless and wired equipment. Base station310 can utilize any equipment suitable for sending and receiving RFtransmissions, such as those employing Code Division Multiple Access(CDMA) communications. In FIG. 3, a mobile radio 350 is shown as well asthe base station 310 including a receive antenna 325, a pair ofreceivers 330 and a multiplexer 340. Additional mobile radios may beserviced by the base station 310, and it will be apparent to one ofordinary skill that base station 310 can be used for providing wirelesscommunications in any desired manner and for any type of wirelesscommunications protocol or standard.

[0046] Numerous modifications and alternative embodiments of theinvention will be apparent to those skilled in the art in view of theforegoing description. Accordingly, this description is to be construedas illustrative only and is for the purpose of teaching those skilled inthe art the best mode of carrying out the invention. Details of thestructure may be varied substantially without departing from the spiritof the invention and the exclusive use of all modifications which comewithin the scope of the appended claim is reserved.

What is claimed is:
 1. An apparatus for modeling and estimating thecharacteristics of a power amplifier, said apparatus comprising: apredistortion module responsive to a predistortion function and an inputsignal by generating a predistorted signal; the amplifier responsive tosaid predistorted signal by generating an output signal; and apolynomial module responsive to said predistorted signal and said outputsignal by generating coefficients of a complex polynomial of order p (pis an integer greater than one), said complex polynomial having botheven and odd terms.
 2. The apparatus according to claim 1, wherein saidpolynomial module employs a minimum mean squared error criteria todetermine said polynomial coefficients.
 3. A wireless radio frequencycommunications system including an apparatus for modeling and estimatingthe characteristics of a power amplifier, said system comprising: apredistortion module responsive to a predistortion function and an inputsignal by generating a predistorted signal; the amplifier responsive tosaid predistorted signal by generating an output signal; and apolynomial module responsive to said predistorted signal and said outputsignal by generating coefficients of a complex polynomial of order p (pis an integer greater than one), said complex polynomial having botheven and odd terms.
 4. The system of claim 3, wherein said polynomialmodule employs a minimum mean squared error criteria to determine saidpolynomial coefficients.
 5. A method for modeling and estimating thecharacteristics of a power amplifier, comprising the steps of:generating a predistorted signal in response to a predistortion functionand an input signal; amplifying said predistorted signal to generate anoutput signal; and generating coefficients of a complex polynomial oforder p (p is an integer greater than one) in response to saidpredistorted signal and said output signal, said complex polynomialincluding both even and odd terms.
 6. The method according to claim 5,wherein said coefficients generating step employs a minimum mean squarederror criteria to determine said polynomial coefficients.